pacman::p_load(sf, raster, spatstat, tmap, tidyverse)2B: 2nd Order Spatial Point Patterns Analysis
In this exercise, we will learn to apply 2nd-order spatial point pattern analysis methods in R, including G, F, K, and L functions, to evaluate spatial point distributions and perform hypothesis testing using the spatstat package.
1 Exercise 2B Reference
R for Geospatial Data Science and Analytics - 5 2nd Order Spatial Point Patterns Analysis Methods
2 Overview
Spatial Point Pattern Analysis is the evaluation of the pattern or distribution, of a set of points on a surface. The point can be location of:
- events such as crime, traffic accident and disease onset, or
- business services (coffee and fastfood outlets) or facilities such as childcare and eldercare.
Using appropriate functions of spatstat, this hands-on exercise aims to discover the spatial point processes of childecare centres in Singapore.
The specific questions we would like to answer are as follows:
- are the childcare centres in Singapore randomly distributed throughout the country?
- if the answer is not, then the next logical question is where are the locations with higher concentration of childcare centres?
This hands-on exercise continues from Hands-on Exercise 2A
Here’s the improved section with relevant information presented in table format:
3 The Data
The following data sources will be used in this exercise:
| Dataset | Description | Source | Format |
|---|---|---|---|
| CHILDCARE | Point data containing location and attributes of childcare centres. | Data.gov.sg | GeoJSON |
| MP14_SUBZONE_WEB_PL | Polygon data representing URA 2014 Master Plan Planning Subzone boundaries. | Data.gov.sg | ESRI Shapefile |
| CoastalOutline | Polygon data showing the national boundary of Singapore. | Singapore Land Authority (SLA) | ESRI Shapefile |
4 Installing and Loading the R Packages
The following R packages will be used in this exercise:
| Package | Purpose | Use Case in Exercise |
|---|---|---|
| sf | Importing, managing, and processing vector-based geospatial data. | Handling vector geospatial data in R. |
| spatstat | Tools for point pattern analysis. | Performing 1st- and 2nd-order spatial point pattern analysis and deriving kernel density estimation (KDE). |
| raster | Reading, writing, manipulating, analyzing, and modeling of gridded spatial data (raster). | Converting image outputs generated by spatstat into raster format. |
| maptools | Tools for manipulating geographic data. | Converting spatial objects into ppp format for use with spatstat. |
| tmap | Functions for plotting cartographic quality static or interactive maps. | Creating static and interactive point pattern maps. |
To install and load these packages in R, use the following code:
5 Spatial Data Wrangling
5.1 Importing the Spatial Data
To import the three geographical datasets, we will use st_read() from sf.
childcare_sf <- st_read("data/child-care-services-geojson.geojson")Reading layer `child-care-services-geojson' from data source
`/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data/child-care-services-geojson.geojson'
using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension: XYZ
Bounding box: xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range: zmin: 0 zmax: 0
Geodetic CRS: WGS 84sg_sf <- st_read(dsn = "data", layer="CostalOutline")Reading layer `CostalOutline' from data source
`/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data'
using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension: XY
Bounding box: xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21mpsz_sf <- st_read(dsn = "data",
layer = "MP14_SUBZONE_WEB_PL")Reading layer `MP14_SUBZONE_WEB_PL' from data source
`/Users/walter/code/isss626/isss626-gaa/Hands-on_Ex/Hands-on_Ex02/data'
using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY215.2 Inspect and Reproject to Same Projection System
Before we can use these data for analysis, it is important for us to ensure that they are projected in same projection system.
First, we check the childcare dataset.
st_crs(childcare_sf)Coordinate Reference System:
User input: WGS 84
wkt:
GEOGCRS["WGS 84",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
CS[ellipsoidal,2],
AXIS["geodetic latitude (Lat)",north,
ORDER[1],
ANGLEUNIT["degree",0.0174532925199433]],
AXIS["geodetic longitude (Lon)",east,
ORDER[2],
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4326]]This dataset is using the WGS84 crs. We will reproject all the dataset to SVY21 crs for standardization and analysis.
childcare_sf <- st_transform(childcare_sf , crs = 3414)
st_crs(childcare_sf)Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]The childcare dataset has been reprojected to SVY21 successfully.
Next, we inspect the Coastal Outline dataset.
st_crs(sg_sf)Coordinate Reference System:
User input: SVY21
wkt:
PROJCRS["SVY21",
BASEGEOGCRS["SVY21[WGS84]",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ID["EPSG",6326]],
PRIMEM["Greenwich",0,
ANGLEUNIT["Degree",0.0174532925199433]]],
CONVERSION["unnamed",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]]]Notice that this dataset is using SVY21 crs, however the ID provided is EPSG:9001 does not match the intended ID, EPSG:3414 of SVY21. In this case, we will set the crs to the correct ID using the code block below.
sg_sf <- st_set_crs(sg_sf, 3414)
st_crs(sg_sf)Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]Similarly, we will inspect the Master Plan Subzone Dataset.
st_crs(mpsz_sf)Coordinate Reference System:
User input: SVY21
wkt:
PROJCRS["SVY21",
BASEGEOGCRS["SVY21[WGS84]",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ID["EPSG",6326]],
PRIMEM["Greenwich",0,
ANGLEUNIT["Degree",0.0174532925199433]]],
CONVERSION["unnamed",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["Degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]]]Since the ID is also EPSG:9001, we will set the crs to EPSG:3414 too.
mpsz_sf <- st_set_crs(mpsz_sf, 3414)
st_crs(mpsz_sf)Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]6 Mapping the Geospatial Datasets
After checking the referencing system of each geospatial data data frame, it is also useful for us to plot a map to show their spatial patterns.
6.1 Static Map
# add polygon layer of the coastal outline of sg island
tm_shape(sg_sf)+ tm_polygons() +
# add polygon layer of the subzone based on sg masterplan
tm_shape(mpsz_sf) + tm_polygons() +
# add dot layer to show the locations of childcare centres
tm_shape(childcare_sf) + tm_dots() +
tm_layout()
When all the 3 datasets are overlayed together, it shows the locations of childcare centres on the Singapore island. Since all the geospatial layers are within the same map context, it means their referencing system and coordinate values are referred to similar spatial context. This consistency is crucial for accurate geospatial analysis.
6.2 Interactive Map
Alternatively, we can also prepare a pin map by using the code block below.
tmap_mode('view')
# tm_basemap("Esri.WorldGrayCanvas") +
# tm_basemap("OpenStreetMap") +
tm_basemap("Esri.WorldTopoMap") +
tm_shape(childcare_sf) +
tm_dots(alpha = 0.5)tmap_mode('plot')In interactive mode, tmap uses the Leaflet for R API, allowing you to freely navigate, zoom, and click on features for detailed information. You can also change the map’s background using layers like ESRI.WorldGrayCanvas, OpenStreetMap, and ESRI.WorldTopoMap, with ESRI.WorldGrayCanvas as the default.
Tip
Remember to switch back to plot mode after interacting to avoid connection issues and limit interactive maps to fewer than 10 in RMarkdown documents for Netlify publishing.
7 Geospatial Data Wrangling
While simple feature data frames are becoming more popular compared to sp’s Spatial* classes, many geospatial analysis packages still require data in the Spatial* format. This section will show you how to convert a simple feature data frame to sp’s Spatial* class.
7.1 Converting sf data frames to sp’s Spatial* class
The code block below uses as_Spatial() of sf package to convert the three geospatial data from simple feature data frame to sp’s Spatial* class.
childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)After the sf dataframe to sp Spatial* conversion, let’s inspect the Spatial* classes.
childcareclass : SpatialPointsDataFrame
features : 1545
extent : 11203.01, 45404.24, 25667.6, 49300.88 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 2
names : Name, Description
min values : kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center>
max values : kml_999, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> mpszclass : SpatialPolygonsDataFrame
features : 323
extent : 2667.538, 56396.44, 15748.72, 50256.33 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 15
names : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C, REGION_N, REGION_C, INC_CRC, FMEL_UPD_D, X_ADDR, Y_ADDR, SHAPE_Leng, SHAPE_Area
min values : 1, 1, ADMIRALTY, AMSZ01, N, ANG MO KIO, AM, CENTRAL REGION, CR, 00F5E30B5C9B7AD8, 16409, 5092.8949, 19579.069, 871.554887798, 39437.9352703
max values : 323, 17, YUNNAN, YSSZ09, Y, YISHUN, YS, WEST REGION, WR, FFCCF172717C2EAF, 16409, 50424.7923, 49552.7904, 68083.9364708, 69748298.792 sgclass : SpatialPolygonsDataFrame
features : 60
extent : 2663.926, 56047.79, 16357.98, 50244.03 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 4
names : GDO_GID, MSLINK, MAPID, COSTAL_NAM
min values : 1, 1, 0, ISLAND LINK
max values : 60, 67, 0, SINGAPORE - MAIN ISLAND The geospatial data have been converted into their respective sp’s Spatial* classes.
7.2 Converting the Generic sp Format into spatstat’s ppp Format
Now, we will use as.ppp() function of spatstat to convert the spatial data into spatstat’s ppp object format.
childcare_ppp <- as.ppp(childcare_sf)
childcare_pppMarked planar point pattern: 1545 points
marks are of storage type 'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] unitsLet’s examine the difference by plotting childcare_ppp:
plot(childcare_ppp)
We can also view the summary statistics of the newly created ppp object by using the code block below.
summary(childcare_ppp)Marked planar point pattern: 1545 points
Average intensity 1.91145e-06 points per square unit
Coordinates are given to 11 decimal places
marks are of type 'character'
Summary:
Length Class Mode
1545 character character
Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
(34200 x 23630 units)
Window area = 808287000 square units
Tip
Be aware of the warning message regarding duplicates. In spatial point pattern analysis, duplicates can be a significant issue. The statistical methods used for analyzing spatial point patterns often assume that the points are distinct and non-coincident.
7.3 Handling Duplicated Points
We can check the duplication in a ppp object by using the duplicated function with different configurations.
Tip
Tips on using duplicated
If rule=“spatstat” (the default), two points are deemed identical if their coordinates are equal according to ==, and their marks are equal according to ==. This is the most stringent possible test. If rule=“unmark”, duplicated points are determined by testing equality of their coordinates only, using ==. If rule=“deldir”, duplicated points are determined by testing equality of their coordinates only, using the function duplicatedxy in the package deldir, which currently uses duplicated.data.frame. Setting rule=“deldir” will ensure consistency with functions in the deldir package.
see R: Determine Duplicated Points in a Spatial Point Pattern
# duplicated(childcare_ppp)
# any(duplicated(childcare_ppp))
rules <- c("spatstat", "deldir", "unmark")
duplicate_counts <- list()
for (rule in rules) {
duplicates <- duplicated(childcare_ppp, rule = rule)
num_duplicates <- sum(duplicates)
duplicate_counts[[rule]] <- num_duplicates
}
print(duplicate_counts)$spatstat
[1] 0
$deldir
[1] 0
$unmark
[1] 74Using unmark, we can find 74 duplicates.
We can also use anyDuplicated(x) as it is a faster version of any(duplicated(x)).
sum(anyDuplicated(childcare_ppp, rule="unmark"))[1] 0To count the number of coincident points, we will use the multiplicity() function as shown in the code block below. see R: Multiplicity for more info.
multiplicity(childcare_ppp)If we want to know how many locations have more than one point event:
sum(multiplicity(childcare_ppp) > 1)[1] 0# double check
coincident_points <- duplicated(childcare_ppp, rule="deldir")
coincident_coordinates <- childcare_ppp[coincident_points]
print(coincident_coordinates)Marked planar point pattern: 0 points
marks are of storage type 'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] unitsThe output shows that there are 128(???) 0 duplicated point events.
tmap_mode('view')
tm_basemap("Esri.WorldTopoMap") +
tm_shape(childcare) +
tm_dots(alpha=0.4,
size=0.05)tmap_mode('plot')7.4 How to Spot Duplicate Points on the Map
There are three ways to overcome this problem. The easiest way is to delete the duplicates. But, that will also mean that some useful point events will be lost.
The second solution is use jittering, which will add a small perturbation to the duplicate points so that they do not occupy the exact same space.
The third solution is to make each point “unique” and then attach the duplicates of the points to the patterns as marks, as attributes of the points. Then you would need analytical techniques that take into account these marks.
7.4.1 Jittering
The code block below implements the jittering approach.
childcare_ppp_jit <- rjitter(childcare_ppp,
retry=TRUE,
nsim=1,
drop=TRUE)any(duplicated(childcare_ppp_jit))[1] FALSE7.5 Creating owin Object
When analysing spatial point patterns, it is a good practice to confine the analysis with a geographical area like Singapore boundary. In spatstat, an object called owin is specially designed to represent this polygonal region.
The code block below is used to covert sg SpatialPolygon object into owin object of spatstat.
sg_owin <- as.owin(sg_sf)The output object can be displayed by using plot() function:
plot(sg_owin)
And using summary() function of Base R:
summary(sg_owin)Window: polygonal boundary
50 separate polygons (1 hole)
vertices area relative.area
polygon 1 (hole) 30 -7081.18 -9.76e-06
polygon 2 55 82537.90 1.14e-04
polygon 3 90 415092.00 5.72e-04
polygon 4 49 16698.60 2.30e-05
polygon 5 38 24249.20 3.34e-05
polygon 6 976 23344700.00 3.22e-02
polygon 7 721 1927950.00 2.66e-03
polygon 8 1992 9992170.00 1.38e-02
polygon 9 330 1118960.00 1.54e-03
polygon 10 175 925904.00 1.28e-03
polygon 11 115 928394.00 1.28e-03
polygon 12 24 6352.39 8.76e-06
polygon 13 190 202489.00 2.79e-04
polygon 14 37 10170.50 1.40e-05
polygon 15 25 16622.70 2.29e-05
polygon 16 10 2145.07 2.96e-06
polygon 17 66 16184.10 2.23e-05
polygon 18 5195 636837000.00 8.78e-01
polygon 19 76 312332.00 4.31e-04
polygon 20 627 31891300.00 4.40e-02
polygon 21 20 32842.00 4.53e-05
polygon 22 42 55831.70 7.70e-05
polygon 23 67 1313540.00 1.81e-03
polygon 24 734 4690930.00 6.47e-03
polygon 25 16 3194.60 4.40e-06
polygon 26 15 4872.96 6.72e-06
polygon 27 15 4464.20 6.15e-06
polygon 28 14 5466.74 7.54e-06
polygon 29 37 5261.94 7.25e-06
polygon 30 111 662927.00 9.14e-04
polygon 31 69 56313.40 7.76e-05
polygon 32 143 145139.00 2.00e-04
polygon 33 397 2488210.00 3.43e-03
polygon 34 90 115991.00 1.60e-04
polygon 35 98 62682.90 8.64e-05
polygon 36 165 338736.00 4.67e-04
polygon 37 130 94046.50 1.30e-04
polygon 38 93 430642.00 5.94e-04
polygon 39 16 2010.46 2.77e-06
polygon 40 415 3253840.00 4.49e-03
polygon 41 30 10838.20 1.49e-05
polygon 42 53 34400.30 4.74e-05
polygon 43 26 8347.58 1.15e-05
polygon 44 74 58223.40 8.03e-05
polygon 45 327 2169210.00 2.99e-03
polygon 46 177 467446.00 6.44e-04
polygon 47 46 699702.00 9.65e-04
polygon 48 6 16841.00 2.32e-05
polygon 49 13 70087.30 9.66e-05
polygon 50 4 9459.63 1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
(53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.4017.6 Combining Point Events Object and Owin Object
For the last step of geospatial data wrangling, we will extract childcare events that are located within Singapore by using the code block below.
childcareSG_ppp = childcare_ppp[sg_owin]The output object combined both the point and polygon feature in one ppp object class as shown below.
summary(childcareSG_ppp)Marked planar point pattern: 1545 points
Average intensity 2.129929e-06 points per square unit
Coordinates are given to 11 decimal places
marks are of type 'character'
Summary:
Length Class Mode
1545 character character
Window: polygonal boundary
50 separate polygons (1 hole)
vertices area relative.area
polygon 1 (hole) 30 -7081.18 -9.76e-06
polygon 2 55 82537.90 1.14e-04
polygon 3 90 415092.00 5.72e-04
polygon 4 49 16698.60 2.30e-05
polygon 5 38 24249.20 3.34e-05
polygon 6 976 23344700.00 3.22e-02
polygon 7 721 1927950.00 2.66e-03
polygon 8 1992 9992170.00 1.38e-02
polygon 9 330 1118960.00 1.54e-03
polygon 10 175 925904.00 1.28e-03
polygon 11 115 928394.00 1.28e-03
polygon 12 24 6352.39 8.76e-06
polygon 13 190 202489.00 2.79e-04
polygon 14 37 10170.50 1.40e-05
polygon 15 25 16622.70 2.29e-05
polygon 16 10 2145.07 2.96e-06
polygon 17 66 16184.10 2.23e-05
polygon 18 5195 636837000.00 8.78e-01
polygon 19 76 312332.00 4.31e-04
polygon 20 627 31891300.00 4.40e-02
polygon 21 20 32842.00 4.53e-05
polygon 22 42 55831.70 7.70e-05
polygon 23 67 1313540.00 1.81e-03
polygon 24 734 4690930.00 6.47e-03
polygon 25 16 3194.60 4.40e-06
polygon 26 15 4872.96 6.72e-06
polygon 27 15 4464.20 6.15e-06
polygon 28 14 5466.74 7.54e-06
polygon 29 37 5261.94 7.25e-06
polygon 30 111 662927.00 9.14e-04
polygon 31 69 56313.40 7.76e-05
polygon 32 143 145139.00 2.00e-04
polygon 33 397 2488210.00 3.43e-03
polygon 34 90 115991.00 1.60e-04
polygon 35 98 62682.90 8.64e-05
polygon 36 165 338736.00 4.67e-04
polygon 37 130 94046.50 1.30e-04
polygon 38 93 430642.00 5.94e-04
polygon 39 16 2010.46 2.77e-06
polygon 40 415 3253840.00 4.49e-03
polygon 41 30 10838.20 1.49e-05
polygon 42 53 34400.30 4.74e-05
polygon 43 26 8347.58 1.15e-05
polygon 44 74 58223.40 8.03e-05
polygon 45 327 2169210.00 2.99e-03
polygon 46 177 467446.00 6.44e-04
polygon 47 46 699702.00 9.65e-04
polygon 48 6 16841.00 2.32e-05
polygon 49 13 70087.30 9.66e-05
polygon 50 4 9459.63 1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
(53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401plot(childcareSG_ppp)
7.6.1 Extracting Study Area
The code block below will be used to extract the target planning areas.
pg <- mpsz_sf %>%
filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
filter(PLN_AREA_N == "JURONG WEST")Plotting the target planning areas:
par(mfrow=c(2,2))
plot(pg, main = "Punggol")
plot(tm, main = "Tampines")
plot(ck, main = "Choa Chu Kang")
plot(jw, main = "Jurong West")
7.6.2 Converting sf objects into owin objects
Now, we will convert these sf objects into owin objects that is required by spatstat.
pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)7.6.3 Combining Childcare Points and the Study Area
To extract childcare that is within the specific region for analysis, we can use:
childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]Next, rescale() function is used to transform the unit of measurement from meter to kilometer.
childcare_pg_ppp.km = rescale(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale(childcare_jw_ppp, 1000, "km")The code block below is used to plot the four study areas and the locations of the childcare centres.
par(mfrow=c(2,2))
plot(childcare_pg_ppp.km, main="Punggol")
plot(childcare_tm_ppp.km, main="Tampines")
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
plot(childcare_jw_ppp.km, main="Jurong West")
8 Second-Order Spatial Point Pattern Analysis
In this section, we will analyze spatial point patterns using various functions: G-Function, F-Function, K-Function, and L-Function.
8.1 Analysing Spatial Point Process Using G-Function
The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, we will learn how to compute G-function estimation by using Gest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.
8.1.1 Choa Chu Kang Planning Area
8.1.1.1 Computing G-function Estimation
To compute G-function using Gest() of spatstat package:
Note
correction is an optional argument in Gest()
Optional. The edge correction(s) to be used to estimate . A vector of character strings selected from “none”, “rs”, “km”, “Hanisch” and “best”. Alternatively correction=“all” selects all options.
# rs and border has the same effect
G_CK = Gest(childcare_ck_ppp, correction = "rs")
G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))
We can also use the “all” option in correction to display all forms of edge corrections from “none”, “rs”, “km”, “Hanisch” and “best”.
G_CK_all = Gest(childcare_ck_ppp, correction = "all")
plot(G_CK_all, xlim=c(0,500))
8.1.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.
\(H_1\)= The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with G-function:
Note
The envelope function calculates overall and pointwise confidence envelopes for a curve based on bootstrap replicates of the curve evaluated at a number of fixed points.
In other words, It helps you determine if your observed spatial pattern (e.g., locations of points in a study area) is significantly different from what you would expect under a random or theoretical distribution.
How It Works: 1. Simulate Data: It generates multiple simulated datasets (often by randomizing the locations of points) based on the null hypothesis (e.g., complete spatial randomness). 2. Compute Statistics: For each simulated dataset, it computes a spatial statistic (e.g., G-function, F-function) and creates a distribution of these statistics. 3. Compare: It compares the observed statistic from your actual data to the distribution of statistics from the simulated datasets. 4. Envelope Plot: It plots the range (envelope) of the simulated statistics along with the observed statistic, allowing you to see if your observed statistic falls outside the range of what is expected under the null hypothesis.
When to Use It? Use the envelope function when you want to:
- Test if the observed spatial pattern deviates significantly from a random pattern or other theoretical patterns.
- Assess the statistical significance of spatial features or clustering in your data.
G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(G_CK.csr)
8.1.2 Tampines Planning Area
8.1.2.1 Computing G-function Estimation
We will use the best edge correction for this example.
G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)
G_tm_all = Gest(childcare_tm_ppp, correction = "all")
plot(G_tm_all)
8.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.
\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
The code block below is used to perform the hypothesis testing.
G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(G_tm.csr)
8.3 Analysing Spatial Point Process Using F-Function
The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape. In this section, we will learn how to compute F-function estimation by using Fest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.
8.3.1 Choa Chu Kang Planning Area
8.3.1.1 Computing F-function estimation
Note
Fest() has the same correction option as Gest().
F_CK_all = Fest(childcare_ck_ppp, correction = "all")
plot(F_CK_all)
8.3.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.
\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
Monte Carlo test with F-function:
F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(F_CK.csr)
8.3.2 Tampines Planning Area
8.3.2.1 Computing F-function estimation
Monte Carlo test with F-function:
F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)
8.3.2.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.
\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.
The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.
The code block below is used to perform the hypothesis testing.
F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)Generating 999 simulations of CSR ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.
Done.plot(F_tm.csr)
8.4 Analysing Spatial Point Process Using K-Function
K-function measures the number of events found up to a given distance of any particular event. In this section, you will learn how to compute K-function estimates by using Kest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.
Note
Kest()’s correction option is different fromGest()andFest()`.
correction: Optional. A character vector containing any selection of the options “none”, “border”, “bord.modif”, “isotropic”, “Ripley”, “translate”, “translation”, “rigid”, “none”, “good” or “best”. It specifies the edge correction(s) to be applied. Alternatively correction=“all” selects all options.
8.4.1 Choa Chu Kang Planning Area
8.4.1.1 Computing K-Function Estimate
K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")
8.4.1.2 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.
\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
The code block below is used to perform the hypothesis testing.
K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")
8.4.2 Tampines Planning Area
8.4.2.1 Computing K-function Estimation
K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r,
ylab= "K(d)-r", xlab = "d(m)",
xlim=c(0,1000))
8.5 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.
\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
The code block below is used to perform the hypothesis testing.
K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(K_tm.csr, . - r ~ r,
xlab="d", ylab="K(d)-r", xlim=c(0,500))
8.6 Analysing Spatial Point Process Using L-Function
In this section, you will learn how to compute L-function estimation by using Lest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.
8.6.1 Choa Chu Kang Planning Area
8.6.1.1 Computing L-function Estimation
Note
Lest() has the same correction options as Kest().
L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r,
ylab= "L(d)-r", xlab = "d(m)")
8.6.1.2 Performing Complete Spatial Randomness Test
\(H_0\) = The distribution of childcare services at Choa Chu Kang are randomly distributed.
\(H_1\) = The distribution of childcare services at Choa Chu Kang are not randomly distributed.
The null hypothesis will be rejected if p-value if smaller than alpha value of 0.001.
The code block below is used to perform the hypothesis testing.
L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")
8.6.2 Tampines Planning Area
8.6.2.1 Computing L-function Estimate
L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r,
ylab= "L(d)-r", xlab = "d(m)",
xlim=c(0,1000))
8.7 Performing Complete Spatial Randomness Test
To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:
\(H_0\) = The distribution of childcare services at Tampines are randomly distributed.
\(H_1\) = The distribution of childcare services at Tampines are not randomly distributed.
The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.
The code chunk below will be used to perform the hypothesis testing
L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)Generating 99 simulations of CSR ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98,
99.
Done.plot(L_tm.csr, . - r ~ r,
xlab="d", ylab="L(d)-r", xlim=c(0,500))